Bayesian GLM Part3

Preparations

Load the necessary libraries

library(rstanarm)   #for fitting models in STAN
library(brms)       #for fitting models in STAN
library(coda)       #for diagnostics
library(bayesplot)  #for diagnostics
library(ggmcmc)     #for MCMC diagnostics
library(DHARMa)     #for residual diagnostics
library(rstan)      #for interfacing with STAN
library(emmeans)    #for marginal means etc
library(broom)      #for tidying outputs
library(tidybayes)  #for more tidying outputs
library(ggeffects)  #for partial plots
library(tidyverse)  #for data wrangling etc
library(broom.mixed)#for summarising models
library(ggeffects)  #for partial effects plots
theme_set(theme_grey()) #put the default ggplot theme back

Scenario

Here is a modified example from Peake and Quinn (1993). Peake and Quinn (1993) investigated the relationship between the number of individuals of invertebrates living in amongst clumps of mussels on a rocky intertidal shore and the area of those mussel clumps.

Format of peakquinn.csv data files

AREA	INDIV
516.00	18
469.06	60
462.25	57
938.60	100
1357.15	48
...	...

AREA	Area of mussel clump mm2 - Predictor variable
INDIV	Number of individuals found within clump - Response variable

The aim of the analysis is to investigate the relationship between mussel clump area and the number of non-mussel invertebrate individuals supported in the mussel clump.

Read in the data

peake = read_csv('../public/data/peakquinn.csv', trim_ws=TRUE)
glimpse(peake)

## Rows: 25
## Columns: 2
## $ AREA  <dbl> 516.00, 469.06, 462.25, 938.60, 1357.15, 1773.66, 1686.01, 1786…
## $ INDIV <dbl> 18, 60, 57, 100, 48, 118, 148, 214, 225, 283, 380, 278, 338, 27…

Exploratory data analysis

When exploring these data as part of a frequentist analysis, exploratory data analysis revealed that the both the response (counds of individuals) and predictor (mussel clump area) were skewed and the relationship between raw counds and mussel clump area was not linear. Furthermore, there was strong evidence of a relationship between mean and variance. Normalising both reponse and predictor addressed these issues. However, rather than log transform the response, it was considered more appropriate to model against a distribution that used a logarithmic link function.

The individual observations here (\(y_i\)) are the observed number of (non mussel individuals found in mussel clump \(i\). As a count, these might be expected to follow a Poisson (or perhaps negative binomial) distribution. In the case of a negative binomial, the observed count for any given mussel clump area are expected to be drawn from a negative binomial distribution with a mean of \(\lambda_i\). All the negative binomial distributions are expected to share the same degree of dispersion (\(\theta\)) - that is, the degree to which the inhabitants of mussell clumps aggregate together (or any other reason for overdispersion) is independent of mussel clump area and can be estimated as a constant across all populations.

The natural log of the expected values (\(\lambda_i\)) is modelled against a linear predictor that includes an intercept (\(\beta_0\)) and a slope (\(\beta_1\)) associated with the natural log of mussel area.

The priors on \(\beta_0\) and \(\beta_1\) should be on the natura log scale (since this will be the scale of the parameters). As starting points, we will consider the following priors:

• \(\beta_0\): Normal prior centered at 0 with a variance of 5
• \(\beta_1\): Normal prior centered at 0 with a variance of 2
• \(\theta\): Exponential prior with rate 1

Model formula: \[ \begin{align} y_i &\sim{} \mathcal{NB}(\lambda_i, \theta)\\ ln(\lambda_i) &= \beta_0 + \beta_1 ln(x_i)\\ \beta_0 & \sim\mathcal{N}(0,5)\\ \beta_1 & \sim\mathcal{N}(0,2)\\ \theta &\sim{} \mathcal{Exp}(1) \end{align} \]

Fit the model

frequentist

summary(glm(INDIV ~ log(AREA), data=peake, family=poisson()))

## 
## Call:
## glm(formula = INDIV ~ log(AREA), family = poisson(), data = peake)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -16.717   -3.614   -1.679    1.808   21.744  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept) 0.013528   0.091147   0.148    0.882    
## log(AREA)   0.691628   0.009866  70.100   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for poisson family taken to be 1)
## 
##     Null deviance: 7494.5  on 24  degrees of freedom
## Residual deviance: 1547.8  on 23  degrees of freedom
## AIC: 1738.9
## 
## Number of Fisher Scoring iterations: 4

summary(MASS::glm.nb(INDIV ~ log(AREA), data=peake))

## 
## Call:
## MASS::glm.nb(formula = INDIV ~ log(AREA), data = peake, init.theta = 7.369263003, 
##     link = log)
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -2.29133  -0.59599  -0.06927   0.48929   1.64734  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -1.16452    0.53387  -2.181   0.0292 *  
## log(AREA)    0.82469    0.06295  13.102   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for Negative Binomial(7.3693) family taken to be 1)
## 
##     Null deviance: 161.076  on 24  degrees of freedom
## Residual deviance:  25.941  on 23  degrees of freedom
## AIC: 312.16
## 
## Number of Fisher Scoring iterations: 1
## 
## 
##               Theta:  7.37 
##           Std. Err.:  2.16 
## 
##  2 x log-likelihood:  -306.16

rstanarm

Using default priors

In rstanarm, the default priors are designed to be weakly informative. They are chosen to provide moderate regularlization (to help prevent overfitting) and help stabalise the computations.

peake.rstanarm = stan_glm(INDIV ~ log(AREA), data=peake,
                          family=poisson(), 
                         iter = 5000, warmup = 1000,
                         chains = 3, thin = 5, refresh = 0)

prior_summary(peake.rstanarm)

## Priors for model 'peake.rstanarm' 
## ------
## Intercept (after predictors centered)
##  ~ normal(location = 0, scale = 2.5)
## 
## Coefficients
##   Specified prior:
##     ~ normal(location = 0, scale = 2.5)
##   Adjusted prior:
##     ~ normal(location = 0, scale = 2)
## ------
## See help('prior_summary.stanreg') for more details

This tells us:

• for the intercept, it is using a normal prior with a mean of 0 and a standard deviation of 2.5. These are the defaults for all non-Gaussian intercepts.

• for the coefficients (in this case, just the slope), the default prior is a normal prior centered around 0 with a standard deviation of 2.5. For Poisson models, this is then adjusted for the scale of the data by dividing the 2.5 by the standard deviation of the (in this case log) predictor (then rounded).

2.5/sd(log(peake$AREA))

## [1] 2.025039

• there is no auxillary parameter and thus there is no auxillary prior

Assessing priors

One way to assess the priors is to have the MCMC sampler sample purely from the prior predictive distribution without conditioning on the observed data. Doing so provides a glimpse at the range of predictions possible under the priors. On the one hand, wide ranging preditions would ensure that the priors are unlikely to influence the actual preditions once they are conditioned on the data. On the other hand, if they are too wide, the sampler is being permitted to traverse into regions of parameter space that are not logically possible in the context of the actual underlying ecological context. Not only could this mean that illogical parameter estimates are possible, when the sampler is traversing regions of paramter space that are not supported by the actual data, the sampler can become unstable and have difficulty.

We can draw from the prior predictive distribution instead of conditioning on the response, by updating the model and indicating prior_PD=TRUE. After refittin the model in this way, we can plot the predictions to gain insights into the range of predictions supported by the priors alone.

peake.rstanarm1 <- update(peake.rstanarm,  prior_PD=TRUE)

ggemmeans(peake.rstanarm1,  ~AREA) %>% plot(add.data=TRUE)

Conclusions:

• we see that the range of predictions are very wide and the slope could range from strongly negative to strongly positive.

Defining priors

The following link provides some guidance about defining priors. [https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations]

When defining our own priors, we typically do not want them to be scaled.

If we wanted to define our own priors that were less vague, yet still not likely to bias the outcomes, we could try the following priors (mainly plucket out of thin air):

• \(\beta_0\): normal centered at 0 with a standard deviation of 5
• \(\beta_1\): normal centered at 0 with a standard deviation of 2

I will also overlay the raw data for comparison.

peake.rstanarm2= stan_glm(INDIV ~ log(AREA), data=peake,
                          family=poisson(), 
                          prior_intercept = normal(0, 5, autoscale=FALSE),
                          prior = normal(0, 2, autoscale=FALSE),
                          prior_PD=TRUE, 
                          iter = 5000, warmup = 1000,
                          chains = 3, thin = 5, refresh = 0
                          )

ggemmeans(peake.rstanarm2,  ~AREA) %>%
  plot(add.data=TRUE)

Now lets refit, conditioning on the data.

peake.rstanarm3= update(peake.rstanarm2,  prior_PD=FALSE)  

Plotting prior and posterior

posterior_vs_prior(peake.rstanarm3, color_by='vs', group_by=TRUE,
                   facet_args=list(scales='free_y'))

Conclusions:

• in each case, the prior is substantially wider than the posterior, suggesting that the posterior is not biased towards the prior.

ggemmeans(peake.rstanarm3,  ~AREA) %>% plot(add.data=TRUE)

brms

Using default priors

In brms, the default priors are designed to be weakly informative. They are chosen to provide moderate regularlization (to help prevent overfitting) and help stabalise the computations.

Unlike rstanarm, brms models must be compiled before they start sampling. For most models, the compilation of the stan code takes around 45 seconds.

peake.brm = brm(bf(INDIV ~ log(AREA), family=poisson()),
                data=peake,
                iter = 5000, warmup = 1000,
                chains = 3, thin = 5, refresh = 0)

prior_summary(peake.brm)

##                   prior     class    coef group resp dpar nlpar bound       source
##                  (flat)         b                                          default
##                  (flat)         b logAREA                             (vectorized)
##  student_t(3, 5.8, 2.5) Intercept                                          default

This tells us:

• for the intercept, it is using a student t (flatter normal) prior with a mean of 5.8 and a standard deviation of 2.5. The mean of this prior is based on the median of the log-transformed response and the standard deviation of 2.5 is the default minimum.

## [1] 5.823046

mad(log(peake$INDIV))

## [1] 1.025465

• for the beta coefficients (in this case, just the slope), the default prior is a inproper flat prior. A flat prior essentially means that any value between negative infinity and positive infinity are equally likely. Whilst this might seem reckless, in practice, it seems to work reasonably well for non-intercept beta parameters.

• there is no sigma parameter in a binomial model and therefore there are no additional priors.

Assessing priors

One way to assess the priors is to have the MCMC sampler sample purely from the prior predictive distribution without conditioning on the observed data. Doing so provides a glimpse at the range of predictions possible under the priors. On the one hand, wide ranging preditions would ensure that the priors are unlikely to influence the actual preditions once they are conditioned on the data. On the other hand, if they are too wide, the sampler is being permitted to traverse into regions of parameter space that are not logically possible in the context of the actual underlying ecological context. Not only could this mean that illogical parameter estimates are possible, when the sampler is traversing regions of paramter space that are not supported by the actual data, the sampler can become unstable and have difficulty.

In brms, we can inform the sampler to draw from the prior predictive distribution instead of conditioning on the response, by running the model with the sample_prior='only' argument. Unfortunately, this cannot be applied when there are flat priors (since the posteriors will necessarily extend to negative and positive infinity). Therefore, in order to use this useful routine, we need to make sure that we have defined a proper prior for all parameters.

peake.brm1 = brm(bf(INDIV ~ log(AREA), family=poisson()),
                 data=peake,
                prior=c(
                  prior(normal(0, 2), class='b')), 
                sample_prior = 'only', 
                iter = 5000, warmup = 1000,
                chains = 3, thin = 5, refresh = 0)

ggemmeans(peake.brm1,  ~AREA) %>% plot(add.data=TRUE)

conditional_effects(peake.brm1) %>%  plot(points=TRUE)

Conclusions:

• we see that the range of predictions is faily wide and the slope could range from strongly negative to strongly positive.

Defining priors

The following link provides some guidance about defining priors. [https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations]

When defining our own priors, we typically do not want them to be scaled.

If we wanted to define our own priors that were less vague, yet still not likely to bias the outcomes, we could try the following priors (mainly plucket out of thin air):

• \(\beta_0\): normal centered at 0 with a standard deviation of 5
• \(\beta_1\): normal centered at 0 with a standard deviation of 2

I will also overlay the raw data for comparison.

peake.brm2 = brm(bf(INDIV ~ log(AREA), family=poisson()),
                 data=peake,
                 prior=c(
                   prior(normal(0, 5),  class='Intercept'),
                   prior(normal(0, 2), class='b')
                 ), 
                 sample_prior = 'only', 
                 iter = 5000, warmup = 1000,
                 chains = 3, thin = 5, refresh = 0)

ggemmeans(peake.brm2,  ~AREA) %>%
  plot(add.data=TRUE)

peake.brm3 <- update(peake.brm2,  sample_prior=TRUE, refresh=0)

Plotting prior and posterior

peake.brm3 %>% get_variables()

##  [1] "b_Intercept"     "b_logAREA"       "prior_Intercept" "prior_b"        
##  [5] "lp__"            "accept_stat__"   "stepsize__"      "treedepth__"    
##  [9] "n_leapfrog__"    "divergent__"     "energy__"

peake.brm3 %>%
  posterior_samples %>%
  select(-`lp__`) %>%
  gather %>%
  mutate(Type=ifelse(str_detect(key, 'prior'), 'Prior', 'b'),
         Class=ifelse(str_detect(key, 'Intercept'),  'Intercept',
               ifelse(str_detect(key, 'b'),  'b', NA))) %>%
  ggplot(aes(x=Type,  y=value)) +
  stat_pointinterval()+
  facet_wrap(~Class,  scales='free')

Exploring the stan code

standata(peake.brm3)

## $N
## [1] 25
## 
## $Y
##  [1]   18   60   57  100   48  118  148  214  225  283  380  278  338  274  569
## [16]  509  682  600  978  363 1402  675 1159 1062  632
## 
## $K
## [1] 2
## 
## $X
##    Intercept   logAREA
## 1          1  6.246107
## 2          1  6.150731
## 3          1  6.136106
## 4          1  6.844389
## 5          1  7.213142
## 6          1  7.480800
## 7          1  7.430120
## 8          1  7.487896
## 9          1  8.035949
## 10         1  8.289067
## 11         1  8.394989
## 12         1  8.401037
## 13         1  8.513765
## 14         1  8.400853
## 15         1  8.610818
## 16         1  8.919481
## 17         1  8.873303
## 18         1  9.121503
## 19         1  9.223560
## 20         1  9.136445
## 21         1  9.527275
## 22         1  9.918414
## 23         1 10.115073
## 24         1 10.208912
## 25         1 10.170373
## attr(,"assign")
## [1] 0 1
## 
## $prior_only
## [1] 0
## 
## attr(,"class")
## [1] "standata"

stancode(peake.brm3)

## // generated with brms 2.14.0
## functions {
## }
## data {
##   int<lower=1> N;  // total number of observations
##   int Y[N];  // response variable
##   int<lower=1> K;  // number of population-level effects
##   matrix[N, K] X;  // population-level design matrix
##   int prior_only;  // should the likelihood be ignored?
## }
## transformed data {
##   int Kc = K - 1;
##   matrix[N, Kc] Xc;  // centered version of X without an intercept
##   vector[Kc] means_X;  // column means of X before centering
##   for (i in 2:K) {
##     means_X[i - 1] = mean(X[, i]);
##     Xc[, i - 1] = X[, i] - means_X[i - 1];
##   }
## }
## parameters {
##   vector[Kc] b;  // population-level effects
##   real Intercept;  // temporary intercept for centered predictors
## }
## transformed parameters {
## }
## model {
##   // likelihood including all constants
##   if (!prior_only) {
##     target += poisson_log_glm_lpmf(Y | Xc, Intercept, b);
##   }
##   // priors including all constants
##   target += normal_lpdf(b | 0, 2);
##   target += normal_lpdf(Intercept | 0, 5);
## }
## generated quantities {
##   // actual population-level intercept
##   real b_Intercept = Intercept - dot_product(means_X, b);
##   // additionally draw samples from priors
##   real prior_b = normal_rng(0,2);
##   real prior_Intercept = normal_rng(0,5);
## }

MCMC sampling diagnostics

In addition to the regular model diagnostics checking, for Bayesian analyses, it is also necessary to explore the MCMC sampling diagnostics to be sure that the chains are well mixed and have converged on a stable posterior.

There are a wide variety of tests that range from the big picture, overal chain characteristics to the very specific detailed tests that allow the experienced modeller to drill down to the very fine details of the chain behaviour. Furthermore, there are a multitude of packages and approaches for exploring these diagnostics.

rstanarm

bayesplot

The bayesplot package offers a range of MCMC diagnostics as well as Posterior Probability Checks (PPC), all of which have a convenient plot() interface. Lets start with the MCMC diagnostics.

available_mcmc()

## bayesplot MCMC module:
##   mcmc_acf
##   mcmc_acf_bar
##   mcmc_areas
##   mcmc_areas_data
##   mcmc_areas_ridges
##   mcmc_areas_ridges_data
##   mcmc_combo
##   mcmc_dens
##   mcmc_dens_chains
##   mcmc_dens_chains_data
##   mcmc_dens_overlay
##   mcmc_hex
##   mcmc_hist
##   mcmc_hist_by_chain
##   mcmc_intervals
##   mcmc_intervals_data
##   mcmc_neff
##   mcmc_neff_data
##   mcmc_neff_hist
##   mcmc_nuts_acceptance
##   mcmc_nuts_divergence
##   mcmc_nuts_energy
##   mcmc_nuts_stepsize
##   mcmc_nuts_treedepth
##   mcmc_pairs
##   mcmc_parcoord
##   mcmc_parcoord_data
##   mcmc_rank_hist
##   mcmc_rank_overlay
##   mcmc_recover_hist
##   mcmc_recover_intervals
##   mcmc_recover_scatter
##   mcmc_rhat
##   mcmc_rhat_data
##   mcmc_rhat_hist
##   mcmc_scatter
##   mcmc_trace
##   mcmc_trace_data
##   mcmc_trace_highlight
##   mcmc_violin

Of these, we will focus on:

• mcmc_trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different shade of blue, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).

plot(peake.rstanarm3, plotfun='mcmc_trace')

The chains appear well mixed and very similar

• acf (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain

plot(peake.rstanarm3, 'acf_bar')

There is no evidence of autocorrelation in the MCMC samples

• Rhat: Rhat is a measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.

plot(peake.rstanarm3, 'rhat_hist')

All Rhat values are below 1.05, suggesting the chains have converged.

• neff (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).

  If the ratios are low, tightening the priors may help.

plot(peake.rstanarm3, 'neff_hist')

Ratios all very high.

More diagnostics

plot(peake.rstanarm3, 'combo')

plot(peake.rstanarm3, 'violin')

stan plots

The rstan package offers a range of MCMC diagnostics. Lets start with the MCMC diagnostics.

Of these, we will focus on:

• stan_trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).

stan_trace(peake.rstanarm3)

The chains appear well mixed and very similar

• stan_acf (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain

stan_ac(peake.rstanarm3) 

There is no evidence of autocorrelation in the MCMC samples

• stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.

stan_rhat(peake.rstanarm3) 

All Rhat values are below 1.05, suggesting the chains have converged.

• stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).

  If the ratios are low, tightening the priors may help.

stan_ess(peake.rstanarm3)

Ratios all very high.

stan_dens(peake.rstanarm3, separate_chains = TRUE)

ggmcmc

The ggmean package also has a set of MCMC diagnostic functions. Lets start with the MCMC diagnostics.

Of these, we will focus on:

• ggs_traceplot: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).

peake.ggs <- ggs(peake.rstanarm3)
ggs_traceplot(peake.ggs)

The chains appear well mixed and very similar

• gss_autocorrelation (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain

ggs_autocorrelation(peake.ggs)

There is no evidence of autocorrelation in the MCMC samples

• stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.

ggs_Rhat(peake.ggs)

All Rhat values are below 1.05, suggesting the chains have converged.

• stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).

  If the ratios are low, tightening the priors may help.

ggs_effective(peake.ggs)

Ratios all very high.

More diagnostics

ggs_crosscorrelation(peake.ggs)

ggs_grb(peake.ggs)

brms

bayesplot

The bayesplot package offers a range of MCMC diagnostics as well as Posterior Probability Checks (PPC), all of which have a convenient plot() interface. Lets start with the MCMC diagnostics.

available_mcmc()

## bayesplot MCMC module:
##   mcmc_acf
##   mcmc_acf_bar
##   mcmc_areas
##   mcmc_areas_data
##   mcmc_areas_ridges
##   mcmc_areas_ridges_data
##   mcmc_combo
##   mcmc_dens
##   mcmc_dens_chains
##   mcmc_dens_chains_data
##   mcmc_dens_overlay
##   mcmc_hex
##   mcmc_hist
##   mcmc_hist_by_chain
##   mcmc_intervals
##   mcmc_intervals_data
##   mcmc_neff
##   mcmc_neff_data
##   mcmc_neff_hist
##   mcmc_nuts_acceptance
##   mcmc_nuts_divergence
##   mcmc_nuts_energy
##   mcmc_nuts_stepsize
##   mcmc_nuts_treedepth
##   mcmc_pairs
##   mcmc_parcoord
##   mcmc_parcoord_data
##   mcmc_rank_hist
##   mcmc_rank_overlay
##   mcmc_recover_hist
##   mcmc_recover_intervals
##   mcmc_recover_scatter
##   mcmc_rhat
##   mcmc_rhat_data
##   mcmc_rhat_hist
##   mcmc_scatter
##   mcmc_trace
##   mcmc_trace_data
##   mcmc_trace_highlight
##   mcmc_violin

Of these, we will focus on:

• trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different shade of blue, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).

mcmc_plot(peake.brm3, type='trace')

The chains appear well mixed and very similar

• acf_bar (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain

mcmc_plot(peake.brm3, type='acf_bar')

There is no evidence of autocorrelation in the MCMC samples

• rhat_hist: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.

mcmc_plot(peake.brm3, type='rhat_hist')

All Rhat values are below 1.05, suggesting the chains have converged.

• neff_hist (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).

  If the ratios are low, tightening the priors may help.

mcmc_plot(peake.brm3, type='neff_hist')

Ratios all very high.

More diagnostics

mcmc_plot(peake.brm3, type='combo')

mcmc_plot(peake.brm3, type='violin')

stan plots

The rstan package offers a range of MCMC diagnostics. Lets start with the MCMC diagnostics.

Of these, we will focus on:

• stan_trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).

stan_trace(peake.brm3$fit)

The chains appear well mixed and very similar

• stan_acf (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain

stan_ac(peake.brm3$fit) 

There is no evidence of autocorrelation in the MCMC samples

• stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.

stan_rhat(peake.brm3$fit) 

All Rhat values are below 1.05, suggesting the chains have converged.

• stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).

  If the ratios are low, tightening the priors may help.

stan_ess(peake.brm3$fit)

Ratios all very high.

stan_dens(peake.brm3$fit, separate_chains = TRUE)

ggmcmc

The ggmean package also has a set of MCMC diagnostic functions. Lets start with the MCMC diagnostics.

Of these, we will focus on:

• ggs_traceplot: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).

peake.ggs <- ggs(peake.brm3)
ggs_traceplot(peake.ggs)

The chains appear well mixed and very similar

• gss_autocorrelation (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain

ggs_autocorrelation(peake.ggs)

There is no evidence of autocorrelation in the MCMC samples

• stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.

ggs_Rhat(peake.ggs)

All Rhat values are below 1.05, suggesting the chains have converged.

• stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).

  If the ratios are low, tightening the priors may help.

ggs_effective(peake.ggs)

Ratios all very high.

More diagnostics

ggs_crosscorrelation(peake.ggs)

ggs_grb(peake.ggs)

Model validation

rstanarm

pp check

Post predictive checks provide additional diagnostics about the fit of the model. Specifically, they provide a comparison between predictions drawn from the model and the observed data used to train the model.

available_ppc()

## bayesplot PPC module:
##   ppc_bars
##   ppc_bars_grouped
##   ppc_boxplot
##   ppc_data
##   ppc_dens
##   ppc_dens_overlay
##   ppc_ecdf_overlay
##   ppc_error_binned
##   ppc_error_hist
##   ppc_error_hist_grouped
##   ppc_error_scatter
##   ppc_error_scatter_avg
##   ppc_error_scatter_avg_vs_x
##   ppc_freqpoly
##   ppc_freqpoly_grouped
##   ppc_hist
##   ppc_intervals
##   ppc_intervals_data
##   ppc_intervals_grouped
##   ppc_loo_intervals
##   ppc_loo_pit
##   ppc_loo_pit_overlay
##   ppc_loo_pit_qq
##   ppc_loo_ribbon
##   ppc_ribbon
##   ppc_ribbon_data
##   ppc_ribbon_grouped
##   ppc_rootogram
##   ppc_scatter
##   ppc_scatter_avg
##   ppc_scatter_avg_grouped
##   ppc_stat
##   ppc_stat_2d
##   ppc_stat_freqpoly_grouped
##   ppc_stat_grouped
##   ppc_violin_grouped

• dens_overlay: plots the density distribution of the observed data (black line) overlayed ontop of 50 density distributions generated from draws from the model (light blue). Ideally, the 50 realisations should be roughly consistent with the observed data.

pp_check(peake.rstanarm3,  plotfun='dens_overlay')

The model draws appear deviate from the observed data.

• error_scatter_avg: this plots the observed values against the average residuals. Similar to a residual plot, we do not want to see any patterns in this plot. There is some pattern remaining in these residuals.

pp_check(peake.rstanarm3, plotfun='error_scatter_avg')

The predictive error seems to be related to the predictor - the model performs poorest at higher mussel clump areas.

• error_scatter_avg_vs_x: this is similar to a regular residual plot and as such should be interpreted as such. Again, this is not interpretable for binary data.

pp_check(peake.rstanarm3, x=peake$AREA, plotfun='error_scatter_avg_vs_x')

• intervals: plots the observed data overlayed ontop of posterior predictions associated with each level of the predictor. Ideally, the observed data should all fall within the predictive intervals.

pp_check(peake.rstanarm3, x=peake$AREA, plotfun='intervals')

The modelled preditions seem to underestimate the uncertainty with increasing mussel clump area.

• ribbon: this is just an alternative way of expressing the above plot.

pp_check(peake.rstanarm3, x=peake$AREA, plotfun='ribbon')

The shinystan package allows the full suite of MCMC diagnostics and posterior predictive checks to be accessed via a web interface.

#library(shinystan)
#launch_shinystan(peake.rstanarm3)

DHARMa residuals

DHARMa residuals provide very useful diagnostics. Unfortunately, we cannot directly use the simulateResiduals() function to generate the simulated residuals. However, if we are willing to calculate some of the components ourself, we can still obtain the simulated residuals from the fitted stan model.

We need to supply:

• simulated (predicted) responses associated with each observation.
• observed values
• fitted (predicted) responses (averaged) associated with each observation

preds <- posterior_predict(peake.rstanarm3,  nsamples=250,  summary=FALSE)
peake.resids <- createDHARMa(simulatedResponse = t(preds),
                            observedResponse = peake$INDIV,
                            fittedPredictedResponse = apply(preds, 2, median),
                            integerResponse = TRUE)
plot(peake.resids)

Conclusions:

• the simulated residuals suggest a general lack of fit due to overdispersion and outliers
• perhaps we should explore a negative binomial model

brms

pp check

Post predictive checks provide additional diagnostics about the fit of the model. Specifically, they provide a comparison between predictions drawn from the model and the observed data used to train the model.

available_ppc()

## bayesplot PPC module:
##   ppc_bars
##   ppc_bars_grouped
##   ppc_boxplot
##   ppc_data
##   ppc_dens
##   ppc_dens_overlay
##   ppc_ecdf_overlay
##   ppc_error_binned
##   ppc_error_hist
##   ppc_error_hist_grouped
##   ppc_error_scatter
##   ppc_error_scatter_avg
##   ppc_error_scatter_avg_vs_x
##   ppc_freqpoly
##   ppc_freqpoly_grouped
##   ppc_hist
##   ppc_intervals
##   ppc_intervals_data
##   ppc_intervals_grouped
##   ppc_loo_intervals
##   ppc_loo_pit
##   ppc_loo_pit_overlay
##   ppc_loo_pit_qq
##   ppc_loo_ribbon
##   ppc_ribbon
##   ppc_ribbon_data
##   ppc_ribbon_grouped
##   ppc_rootogram
##   ppc_scatter
##   ppc_scatter_avg
##   ppc_scatter_avg_grouped
##   ppc_stat
##   ppc_stat_2d
##   ppc_stat_freqpoly_grouped
##   ppc_stat_grouped
##   ppc_violin_grouped

• dens_overlay: plots the density distribution of the observed data (black line) overlayed ontop of 50 density distributions generated from draws from the model (light blue). Ideally, the 50 realisations should be roughly consistent with the observed data.

pp_check(peake.brm3,  type='dens_overlay')

The model draws appear deviate from the observed data.

• error_scatter_avg: this plots the observed values against the average residuals. Similar to a residual plot, we do not want to see any patterns in this plot. There is some pattern remaining in these residuals.

pp_check(peake.brm3, type='error_scatter_avg')

The predictive error seems to be related to the predictor - the model performs poorest at higher mussel clump areas.

• error_scatter_avg_vs_x: this is similar to a regular residual plot and as such should be interpreted as such. Again, this is not interpretable for binary data.

pp_check(peake.brm3, x='AREA', type='error_scatter_avg_vs_x')

• intervals: plots the observed data overlayed ontop of posterior predictions associated with each level of the predictor. Ideally, the observed data should all fall within the predictive intervals.

pp_check(peake.brm3, x='AREA', type='intervals')

The modelled preditions seem to underestimate the uncertainty with increasing mussel clump area.

• ribbon: this is just an alternative way of expressing the above plot.

pp_check(peake.brm3, x='AREA', type='ribbon')

The shinystan package allows the full suite of MCMC diagnostics and posterior predictive checks to be accessed via a web interface.

#library(shinystan)
#launch_shinystan(peake.brm3)

DHARMa residuals

DHARMa residuals provide very useful diagnostics. Unfortunately, we cannot directly use the simulateResiduals() function to generate the simulated residuals. However, if we are willing to calculate some of the components ourself, we can still obtain the simulated residuals from the fitted stan model.

We need to supply:

• simulated (predicted) responses associated with each observation.
• observed values
• fitted (predicted) responses (averaged) associated with each observation

preds <- posterior_predict(peake.brm3,  nsamples=250,  summary=FALSE)
peake.resids <- createDHARMa(simulatedResponse = t(preds),
                            observedResponse = peake$INDIV,
                            fittedPredictedResponse = apply(preds, 2, median),
                            integerResponse = TRUE)
plot(peake.resids)

Conclusions:

• the simulated residuals suggest a general lack of fit due to overdispersion and outliers
• perhaps we should explore a negative binomial model

Fit Negative Binomial model

rstanarm

Fit the model

peake.rstanarm4= stan_glm(INDIV ~ log(AREA), data=peake,
                          family=neg_binomial_2(), 
                          prior_intercept = normal(0, 5, autoscale=FALSE),
                          prior = normal(0, 2, autoscale=FALSE),
                          prior_aux = rstanarm::exponential(rate=1, autoscale=FALSE), 
                          iter = 5000, warmup = 1000,
                          chains = 3, thin = 5, refresh = 0
                          )

Plotting prior and posterior

posterior_vs_prior(peake.rstanarm4, color_by='vs', group_by=TRUE,
                   facet_args=list(scales='free_y'))

Conclusions:

• in each case, the prior is substantially wider than the posterior, suggesting that the posterior is not biased towards the prior.

ggemmeans(peake.rstanarm4,  ~AREA) %>% plot(add.data=TRUE)

MCMC chain and posterior checks

plot(peake.rstanarm4, plotfun='mcmc_trace')

plot(peake.rstanarm4, 'acf_bar')

plot(peake.rstanarm4, 'rhat_hist')

plot(peake.rstanarm4, 'neff_hist')

pp_check(peake.rstanarm4, x=peake$AREA, plotfun='error_scatter_avg_vs_x')

pp_check(peake.rstanarm4, x=peake$AREA, plotfun='intervals')

DHARMa residuals

preds <- posterior_predict(peake.rstanarm4,  nsamples=250,  summary=FALSE)
peake.resids <- createDHARMa(simulatedResponse = t(preds),
                            observedResponse = peake$INDIV,
                            fittedPredictedResponse = apply(preds, 2, median),
                            integerResponse = TRUE)
plot(peake.resids)

LOOIC

Different models can be compared via information criterion. LOO (Leave One Out) IC is similar to AIC except that AIC does not consider priors and assumes that the posterior likelihood is multivariate normal. LOO AIC does not and integrates over all uncertainty.

• ELPD: is the theoretical expected log pointwise predictive density for a new dataset and can be estimated via leave-one-out cross-validation.
• elpd_loo: is the Bayesian LOO estimate of ELPD
• SE of elpd_loo: is the Monte Carlo standard error is the estimate for the computational accuracy of MCMC and importance sampling used to compute elpd_loo
• p_loo: is the difference between elpd_loo and the estimated from non-cross validated ELPD and therefore is a measure of the relative extra difficulty of predicting new data compared to the observed data. It can also be a meaure of the effective number of parameters:
  • in a well behaved model, p_loo will be less than the number of estimated parameters and the total sample size.
  • in a misspecified model, p_loo will be greater than the number of estimated parameters and the total sample size.
• Pareto k: is a relative measure of the influence of each observation as well as the accuracy with which the associated leave one out calculations can be estimated.
  • when k < 0.5: the corresponding components can be accurately estimated
  • when 0.5 < k < 0.7: the accuracy is lower but still acceptable
  • when k>0.7: the accuracy is too low and elpd_loo is unreliable. This can also suggest a misspecified model.

More information about the above can be gleaned from [https://mc-stan.org/loo/reference/loo-glossary.html].

Start with the Poisson model

(peake.rstanarm3.loo =loo(peake.rstanarm3))

## 
## Computed from 2400 by 25 log-likelihood matrix
## 
##          Estimate    SE
## elpd_loo   -932.9 298.9
## p_loo       122.2  49.0
## looic      1865.9 597.9
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     14    56.0%   869       
##  (0.5, 0.7]   (ok)        2     8.0%   81        
##    (0.7, 1]   (bad)       5    20.0%   20        
##    (1, Inf)   (very bad)  4    16.0%   1         
## See help('pareto-k-diagnostic') for details.

Conclusions:

• p_loo is substantially higher than the number of estimated parameters
• there are some Pareto k values identified as bad or even very bad

Now for the Negative Binomial model

(peake.rstanarm4.loo =loo(peake.rstanarm4))

## 
## Computed from 2400 by 25 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -156.7  5.4
## p_loo         1.9  0.5
## looic       313.4 10.7
## ------
## Monte Carlo SE of elpd_loo is 0.0.
## 
## All Pareto k estimates are good (k < 0.5).
## See help('pareto-k-diagnostic') for details.

Conclusions:

• p_loo is lower than the number of estimated parameters
• there are no bad Pareto k values

We can also compare the models.

The difference in Expected Log Pointwise Probability Density (elpd_dff) computes the difference in elpd_loo of two models (expressed relative to the higher model). The difference in elpd_diff will be negative if the expected out-of-sample predictive accuracy of the first model is higher. If the difference is be positive then the second model is preferred.

Note, the above assumes that both models are valid. In the current case, we know that the Poisson model was overdispersed and thus it should not be a genuine candidate. Nevertheless, we will compare the Poisson to the Negative Binomial for illustrative purposes.

loo_compare(peake.rstanarm3.loo, peake.rstanarm4.loo)

##                 elpd_diff se_diff
## peake.rstanarm4    0.0       0.0 
## peake.rstanarm3 -776.2     295.4

Conclusions:

• the elpd_diff is negative, indicating that the first model (Negative Binomial) is better.

brms

Fit the model

peake.brm4 = brm(bf(INDIV ~ log(AREA), family=negbinomial()),
                 data=peake,
                 prior=c(
                   prior(normal(0, 5),  class='Intercept'),
                   prior(normal(0, 2), class='b'),
                   prior(gamma(0.01, 0.01), class='shape')
                 ),
                 sample_prior=TRUE, 
                 iter = 5000, warmup = 1000,
                 chains = 3, thin = 5, refresh = 0)

Plotting prior and posterior

peake.brm4 %>%
  posterior_samples %>%
  select(-`lp__`) %>%
  gather %>%
  mutate(Type=ifelse(str_detect(key, 'prior'), 'Prior', 'b'),
         Class=ifelse(str_detect(key, 'Intercept'),  'Intercept',
               ifelse(str_detect(key, 'b'),  'b', 'shape'))) %>%
  ggplot(aes(x=Type,  y=value)) +
  stat_pointinterval()+
  facet_wrap(~Class,  scales='free')

Conclusions:

• in each case, the prior is substantially wider than the posterior, suggesting that the posterior is not biased towards the prior.

ggemmeans(peake.brm4,  ~AREA) %>% plot(add.data=TRUE)

MCMC chain and posterior checks

plot(peake.brm4, type='mcmc_trace')

plot(peake.brm4, type='acf_bar')

plot(peake.brm4, type='rhat_hist')

plot(peake.brm4, type='neff_hist')

pp_check(peake.brm4, x='AREA',type='error_scatter_avg_vs_x')

pp_check(peake.brm4, x='AREA', type='intervals')

DHARMa residuals

preds <- posterior_predict(peake.brm4,  nsamples=250,  summary=FALSE)
peake.resids <- createDHARMa(simulatedResponse = t(preds),
                            observedResponse = peake$INDIV,
                            fittedPredictedResponse = apply(preds, 2, median),
                            integerResponse = TRUE)
plot(peake.resids)

LOOIC

Different models can be compared via information criterion. LOO (Leave One Out) IC is similar to AIC except that AIC does not consider priors and assumes that the posterior likelihood is multivariate normal. LOO AIC does not and integrates over all uncertainty.

• ELPD: is the theoretical expected log pointwise predictive density for a new dataset and can be estimated via leave-one-out cross-validation.
• elpd_loo: is the Bayesian LOO estimate of ELPD
• SE of elpd_loo: is the Monte Carlo standard error is the estimate for the computational accuracy of MCMC and importance sampling used to compute elpd_loo
• p_loo: is the difference between elpd_loo and the estimated from non-cross validated ELPD and therefore is a measure of the relative extra difficulty of predicting new data compared to the observed data. It can also be a meaure of the effective number of parameters:
  • in a well behaved model, p_loo will be less than the number of estimated parameters and the total sample size.
  • in a misspecified model, p_loo will be greater than the number of estimated parameters and the total sample size.
• Pareto k: is a relative measure of the influence of each observation as well as the accuracy with which the associated leave one out calculations can be estimated.
  • when k < 0.5: the corresponding components can be accurately estimated
  • when 0.5 < k < 0.7: the accuracy is lower but still acceptable
  • when k>0.7: the accuracy is too low and elpd_loo is unreliable. This can also suggest a misspecified model.

More information about the above can be gleaned from [https://mc-stan.org/loo/reference/loo-glossary.html].

Start with the Poisson model

(peake.brm3.loo =loo(peake.brm3))

## 
## Computed from 2400 by 25 log-likelihood matrix
## 
##          Estimate    SE
## elpd_loo   -928.4 296.9
## p_loo       120.3  48.2
## looic      1856.7 593.8
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     15    60.0%   522       
##  (0.5, 0.7]   (ok)        3    12.0%   73        
##    (0.7, 1]   (bad)       4    16.0%   24        
##    (1, Inf)   (very bad)  3    12.0%   1         
## See help('pareto-k-diagnostic') for details.

Conclusions:

• p_loo is substantially higher than the number of estimated parameters
• there are some Pareto k values identified as bad or even very bad

Now for the Negative Binomial model

(peake.brm4.loo =loo(peake.brm4))

## 
## Computed from 2400 by 25 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -156.4  5.8
## p_loo         3.0  0.9
## looic       312.7 11.5
## ------
## Monte Carlo SE of elpd_loo is 0.1.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     24    96.0%   720       
##  (0.5, 0.7]   (ok)        1     4.0%   1694      
##    (0.7, 1]   (bad)       0     0.0%   <NA>      
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## 
## All Pareto k estimates are ok (k < 0.7).
## See help('pareto-k-diagnostic') for details.

Conclusions:

• p_loo is lower than the number of estimated parameters
• there are no bad Pareto k values

We can also compare the models.

The difference in Expected Log Pointwise Probability Density (elpd_dff) computes the difference in elpd_loo of two models (expressed relative to the higher model). The difference in elpd_diff will be negative if the expected out-of-sample predictive accuracy of the first model is higher. If the difference is be positive then the second model is preferred.

Note, the above assumes that both models are valid. In the current case, we know that the Poisson model was overdispersed and thus it should not be a genuine candidate. Nevertheless, we will compare the Poisson to the Negative Binomial for illustrative purposes.

loo_compare(peake.brm3.loo, peake.brm4.loo)

##            elpd_diff se_diff
## peake.brm4    0.0       0.0 
## peake.brm3 -772.0     292.6

Conclusions:

• the elpd_diff is negative, indicating that the first model (Negative Binomial) is better.

Partial effects plots

rstanarm

ggpredict

peake.rstanarm4 %>% ggpredict() %>% plot(add.data=TRUE)

## $AREA

ggemmeans

peake.rstanarm4 %>% ggemmeans(~AREA,  type='fixed', transform='response') %>% plot(add.data=TRUE)

fitted_draws

peake.rstanarm4 %>% fitted_draws(newdata=peake) %>%
  median_hdci() %>%
  ggplot(aes(x=AREA, y=.value)) +
  geom_ribbon(aes(ymin=.lower, ymax=.upper), fill='blue', alpha=0.3) + 
  geom_line() +
  geom_point(data=peake,  aes(y=INDIV,  x=AREA))

brms

conditional_effects

peake.brm4 %>% conditional_effects() 

peake.brm4 %>% conditional_effects(spaghetti=TRUE,nsamples=200) 

ggpredict

peake.brm4 %>% ggpredict() %>% plot(add.data=TRUE)

## $AREA

ggemmeans

peake.brm4 %>% ggemmeans(~AREA) %>% plot(add.data=TRUE)

fitted_draws

peake.brm4 %>% fitted_draws(newdata=peake) %>%
  median_hdci() %>%
  ggplot(aes(x=AREA, y=.value)) +
  geom_ribbon(aes(ymin=.lower, ymax=.upper), fill='blue', alpha=0.3) + 
  geom_line() +
  geom_point(data=peake,  aes(y=INDIV,  x=AREA))

Model investigation

rstanarm

rstanarm captures the MCMC samples from stan within the returned list. There are numerous ways to retrieve and summarise these samples. The first three provide convenient numeric summaries from which you can draw conclusions, the last four provide ways of obtaining the full posteriors.

summary

The summary() method generates simple summaries (mean, standard deviation as well as 10, 50 and 90 percentiles).

summary(peake.rstanarm4)

## 
## Model Info:
##  function:     stan_glm
##  family:       neg_binomial_2 [log]
##  formula:      INDIV ~ log(AREA)
##  algorithm:    sampling
##  sample:       2400 (posterior sample size)
##  priors:       see help('prior_summary')
##  observations: 25
##  predictors:   2
## 
## Estimates:
##                         mean   sd   10%   50%   90%
## (Intercept)           -1.2    0.7 -2.1  -1.2  -0.2 
## log(AREA)              0.8    0.1  0.7   0.8   0.9 
## reciprocal_dispersion  4.7    1.3  3.1   4.5   6.4 
## 
## Fit Diagnostics:
##            mean   sd    10%   50%   90%
## mean_PPD 479.7   89.0 377.1 471.2 596.1
## 
## The mean_ppd is the sample average posterior predictive distribution of the outcome variable (for details see help('summary.stanreg')).
## 
## MCMC diagnostics
##                       mcse Rhat n_eff
## (Intercept)           0.0  1.0  2458 
## log(AREA)             0.0  1.0  2477 
## reciprocal_dispersion 0.0  1.0  2343 
## mean_PPD              1.7  1.0  2624 
## log-posterior         0.0  1.0  2348 
## 
## For each parameter, mcse is Monte Carlo standard error, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence Rhat=1).

Conclusions:

• in the Model Info, we are informed that the total MCMC posterior sample size is 2400 and that there were 19 raw observations.
• the estimated intercept (expected number of individuals for a mussel clump of log-transformed area = 0) is -1.15. This is the mean of the posterior distribution for this parameter. If we back-transform this to the response scale, this becomes 0.32. When the mussel clump is 0, we expect to find 0.32 individuals in the mussel clump (which obviously does not exist if it has a log-transformed area of 0!).
  
• the estimated slope (rate at which the number of individuals changes per 1 unit change in log-transformed area), is 0.82 (mean) or 0.72 (median) with a standard deviation of 0. The 90% credibility intervals indicate that we are 90% confident that the slope is between 0.82 and 0.82 - e.g. there is a significant positive trend. When back-transformed onto the reponse scale, we see that for a one unit increase in log-transformed mussel clump area, the number of individuals increases by a factor of 2.28. This represents a 128% increase per 1 unit change in log-transformed mussel clump area.
• Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.

tidyMCMC

tidyMCMC(peake.rstanarm4$stanfit, estimate.method='median',  conf.int=TRUE,  conf.method='HPDinterval',  rhat=TRUE, ess=TRUE)

Conclusions:

• the estimated intercept (expected number of individuals for a mussel clump of log-transformed area = 0) is -1.16. This is the median of the posterior distribution for this parameter. If we back-transform this to the response scale, this becomes 0.31. When the mussel clump is 0, we expect to find 0.31 individuals in the mussel clump (which obviously does not exist if it has a log-transformed area of 0!).
  
• the estimated slope (rate at which the number of individuals changes per 1 unit change in log-transformed area), is 0.82 (mean) with a standard error of 0.09. The 95% credibility intervals indicate that we are 90% confident that the slope is between 0.66 and 1 - e.g. there is a significant positive trend. When back-transformed onto the reponse scale, we see that for a one unit increase in log-transformed mussel clump area, the number of individuals increases by a factor of 2.28. This represents a 128% increase per 1 unit change in log-transformed mussel clump area.
• Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.

gather_draws

Due to the presence of a log transform in the predictor, it is better to use the regex version.

peake.rstanarm4 %>% get_variables()

## [1] "(Intercept)"           "log(AREA)"             "reciprocal_dispersion"
## [4] "accept_stat__"         "stepsize__"            "treedepth__"          
## [7] "n_leapfrog__"          "divergent__"           "energy__"

peake.draw <- peake.rstanarm4 %>% gather_draws(`.Intercept.*|.*AREA.*`,  regex=TRUE)
peake.draw

We can then summarise this

peake.draw %>% median_hdci

peake.rstanarm4 %>% 
  gather_draws(`.Intercept.*|.*AREA.*`, regex=TRUE) %>% 
  ggplot() + 
  stat_halfeye(aes(x=.value,  y=.variable)) +
  facet_wrap(~.variable, scales='free')

We could alternatively express the parameters on the response scale.

peake.rstanarm4 %>% 
  gather_draws(`.Intercept.*|.*AREA.*`, regex=TRUE) %>%
  group_by(.variable) %>%
  mutate(.value=exp(.value)) %>%
  median_hdci

Conclusions:

• the estimated intercept (expected number of individuals for a mussel clump of log-transformed area = 0) is -1.16. This is the median of the posterior distribution for this parameter. If we back-transform this to the response scale, this becomes 0.31. When the mussel clump is 0, we expect to find 0.31 individuals in the mussel clump (which obviously does not exist if it has a log-transformed area of 0!).
  
• the estimated slope (rate at which the number of individuals changes per 1 unit change in log-transformed area), is 0.82 (mean) with a standard error of 0.66. The 95% credibility intervals indicate that we are 90% confident that the slope is between 1 and 0.95 - e.g. there is a significant positive trend. When back-transformed onto the reponse scale, we see that for a one unit increase in log-transformed mussel clump area, the number of individuals increases by a factor of 2.28. This represents a 128% increase per 1 unit change in log-transformed mussel clump area.
• Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.

bayesplot

peake.rstanarm4 %>% plot(plotfun='mcmc_intervals') 

tidy_draws

This is purely a graphical depiction on the posteriors.

peake.rstanarm4 %>% tidy_draws()

spread_draws

peake.rstanarm4 %>% spread_draws(`.Intercept.*|.*AREA.*`,  regex=TRUE)

posterior_samples

peake.rstanarm4 %>% posterior_samples() %>% as_tibble()

\(R^2\)

Unfortunately, \(R^2\) calculations for models other than Gaussian and Binomial have not yet been implemented for rstanarm models yet.

#peake.rstanarm4 %>% bayes_R2() %>% median_hdci

brms

brms captures the MCMC samples from stan within the returned list. There are numerous ways to retrieve and summarise these samples. The first three provide convenient numeric summaries from which you can draw conclusions, the last four provide ways of obtaining the full posteriors.

summary

The summary() method generates simple summaries (mean, standard deviation as well as 10, 50 and 90 percentiles).

summary(peake.brm4)

##  Family: negbinomial 
##   Links: mu = log; shape = identity 
## Formula: INDIV ~ log(AREA) 
##    Data: peake (Number of observations: 25) 
## Samples: 3 chains, each with iter = 5000; warmup = 1000; thin = 5;
##          total post-warmup samples = 2400
## 
## Population-Level Effects: 
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept    -1.15      0.61    -2.39     0.03 1.00     2165     2033
## logAREA       0.82      0.07     0.68     0.97 1.00     2162     2198
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## shape     6.85      2.12     3.53    11.43 1.00     2217     2315
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Conclusions:

• in the Model Info, we are informed that the total MCMC posterior sample size is 2400 and that there were 19 raw observations.
• the estimated intercept (expected number of individuals for a mussel clump of log-transformed area = 0) is -1.15. This is the mean of the posterior distribution for this parameter. If we back-transform this to the response scale, this becomes 0.32. When the mussel clump is 0, we expect to find 0.32 individuals in the mussel clump (which obviously does not exist if it has a log-transformed area of 0!).
  
• the estimated slope (rate at which the number of individuals changes per 1 unit change in log-transformed area), is 0.82 (mean) or 0.97 (median) with a standard deviation of 0.07. The 90% credibility intervals indicate that we are 90% confident that the slope is between 0.82 and 1 - e.g. there is a significant positive trend. When back-transformed onto the reponse scale, we see that for a one unit increase in log-transformed mussel clump area, the number of individuals increases by a factor of 2.28. This represents a 128% increase per 1 unit change in log-transformed mussel clump area.
• Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.

tidyMCMC

tidyMCMC(peake.brm4$fit, estimate.method='median',  conf.int=TRUE,  conf.method='HPDinterval',  rhat=TRUE, ess=TRUE)

Conclusions:

• the estimated intercept (expected number of individuals for a mussel clump of log-transformed area = 0) is -1.16. This is the median of the posterior distribution for this parameter. If we back-transform this to the response scale, this becomes 0.31. When the mussel clump is 0, we expect to find 0.31 individuals in the mussel clump (which obviously does not exist if it has a log-transformed area of 0!).
  
• the estimated slope (rate at which the number of individuals changes per 1 unit change in log-transformed area), is 0.82 (mean) with a standard error of 0.07. The 95% credibility intervals indicate that we are 90% confident that the slope is between 0.69 and 0.97 - e.g. there is a significant positive trend. When back-transformed onto the reponse scale, we see that for a one unit increase in log-transformed mussel clump area, the number of individuals increases by a factor of 2.28. This represents a 128% increase per 1 unit change in log-transformed mussel clump area.
• Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.

gather_draws

Due to the presence of a log transform in the predictor, it is better to use the regex version.

peake.brm4 %>% get_variables()

##  [1] "b_Intercept"     "b_logAREA"       "shape"           "prior_Intercept"
##  [5] "prior_b"         "prior_shape"     "lp__"            "accept_stat__"  
##  [9] "stepsize__"      "treedepth__"     "n_leapfrog__"    "divergent__"    
## [13] "energy__"

peake.draw <- peake.brm4 %>% gather_draws(`b_.*`,  regex=TRUE)
peake.draw

We can then summarise this

peake.draw %>% median_hdci

peake.brm4 %>% 
  gather_draws(`b_.*`, regex=TRUE) %>% 
  ggplot() + 
  stat_halfeye(aes(x=.value,  y=.variable)) +
  facet_wrap(~.variable, scales='free')

We could alternatively express the parameters on the response scale.

peake.brm4 %>% 
  gather_draws(`.Intercept.*|.*AREA.*`, regex=TRUE) %>%
  group_by(.variable) %>%
  mutate(.value=exp(.value)) %>%
  median_hdci

Conclusions:

• the estimated intercept (expected number of individuals for a mussel clump of log-transformed area = 0) is -1.16. This is the median of the posterior distribution for this parameter. If we back-transform this to the response scale, this becomes 0.31. When the mussel clump is 0, we expect to find 0.31 individuals in the mussel clump (which obviously does not exist if it has a log-transformed area of 0!).
  
• the estimated slope (rate at which the number of individuals changes per 1 unit change in log-transformed area), is 0.82 (mean) with a standard error of 0.69. The 95% credibility intervals indicate that we are 90% confident that the slope is between 0.97 and 0.95 - e.g. there is a significant positive trend. When back-transformed onto the reponse scale, we see that for a one unit increase in log-transformed mussel clump area, the number of individuals increases by a factor of 2.28. This represents a 128% increase per 1 unit change in log-transformed mussel clump area.
• Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.

bayesplot

peake.brm4 %>% plot(plotfun='mcmc_intervals') 

tidy_draws

This is purely a graphical depiction on the posteriors.

peake.brm4 %>% tidy_draws()

spread_draws

peake.brm4 %>% spread_draws(`b_.*`,  regex=TRUE)

posterior_samples

peake.brm4 %>% posterior_samples() %>% as_tibble()

\(R^2\)

peake.brm4 %>% bayes_R2(summary=FALSE) %>% median_hdci

Hypothesis testing

Since we have the entire posterior, we are able to make probablity statements. We simply count up the number of MCMC sample draws that satisfy a condition (e.g represent a slope greater than 0) and then divide by the total number of MCMC samples.

For this exersize, we will explore the following:

• what this the probability that the number of individuals increases with increasing mussel clump area - this is just asking what proportion of the drawn slopes are greater than 0.
• how much of a change in individuals do we expect if the mussel clump area is increased from 5,000 to 10,000 units and what is the probability that the change is more than 50%

rstanarm

Prob. of effect

Unfortunately, the inline log-transformation of the predictor interfers with hypothesis()’s ability to find the slope. We can get around this by renaming before calling hypothesis().

peake.rstanarm4 %>% as.data.frame() %>% rename(lAREA=`log(AREA)`) %>% hypothesis('lAREA>0')

## Hypothesis Tests for class :
##    Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio Post.Prob Star
## 1 (lAREA) > 0     0.82      0.09     0.69     0.96        Inf         1    *
## ---
## 'CI': 90%-CI for one-sided and 95%-CI for two-sided hypotheses.
## '*': For one-sided hypotheses, the posterior probability exceeds 95%;
## for two-sided hypotheses, the value tested against lies outside the 95%-CI.
## Posterior probabilities of point hypotheses assume equal prior probabilities.

Conclusions:

• the probability of a positive effect of mussel clump area on number of individuals is 1. That is, there is, we are 100% confident that there is an effect.
• the evidence ratio is normally the ratio of the number of cases that satisty the hypothesis to the number of cases that do not. Since the number of cases that do not satisfy the hypothesis is 0, the evidence ratio is Inf (since division by 0)

Alternatively…

peake.rstanarm4 %>% tidy_draws() %>% summarise(P=sum(`log(AREA)`>0)/n())

## # A tibble: 1 x 1
##       P
##   <dbl>
## 1     1

Conclusions:

• the probability of a positive effect of mussel clump area on number of individuals is 1. That is, there is, we are 100% confident that there is an effect.

Change 5000 to 10000

newdata = list(AREA=c(5000, 10000)) 
peake.rstanarm4 %>% emmeans(~AREA,  at=newdata) %>% pairs()

##  contrast     estimate lower.HPD upper.HPD
##  5000 - 10000   -0.571    -0.691    -0.459
## 
## Point estimate displayed: median 
## Results are given on the log (not the response) scale. 
## HPD interval probability: 0.95

Conclusions:

• the change in expected number of individuals associated with an increase in mussel clump area from 5,000 to 10,000 is -0.57.

If we want to derive other properties, such as the percentage change, then we use tidy_draws() and then simple tidyverse spreadsheet operation.

peake.mcmc <- peake.rstanarm4 %>% emmeans(~AREA,  at=newdata) %>% 
  tidy_draws() %>%
  rename_with(~str_replace(., 'AREA ', 'p')) %>%
  mutate(Eff=p10000 - p5000,
         PEff=100*Eff/p5000)
peake.mcmc %>% head

Now we can calculate the medians and HPD intervals of each column (and ignore the .chain, .iteration and .draw).

peake.mcmc %>% tidyMCMC(estimate.method='median',
                       conf.int=TRUE, conf.method='HPDinterval')

Alternatively, we could use median_hdci

peake.mcmc %>% median_hdci(PEff)

Conclusions:

• the percentage change in expected number of individuals associated with an increase in mussel clump area from 5,000 to 10,000 is 9.74% with a 95% credibility interval of 7.71 11.62

To get the probability that the effect is greater than a 50% increase.

peake.mcmc %>% summarise(P=sum(PEff>50)/n())

## # A tibble: 1 x 1
##       P
##   <dbl>
## 1     0

Conclusions:

• the probability that the number of individuals will increase by more than 50% following an increase in mussel clump area from 5,000 to 10,000 is

Finally, we could alternatively use hypothesis(). Note that when we do so, the estimate is the difference between the effect and the hypothesised effect (50%).

peake.mcmc %>% hypothesis('PEff>50')

## Hypothesis Tests for class :
##        Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio Post.Prob
## 1 (PEff)-(50) > 0   -40.26         1   -41.92   -38.66          0         0
##   Star
## 1     
## ---
## 'CI': 90%-CI for one-sided and 95%-CI for two-sided hypotheses.
## '*': For one-sided hypotheses, the posterior probability exceeds 95%;
## for two-sided hypotheses, the value tested against lies outside the 95%-CI.
## Posterior probabilities of point hypotheses assume equal prior probabilities.

brm

Prob. of effect

Unfortunately, the inline log-transformation of the predictor interfers with hypothesis()’s ability to find the slope.

peake.brm4 %>% hypothesis('logAREA>0')

## Hypothesis Tests for class b:
##      Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio Post.Prob Star
## 1 (logAREA) > 0     0.82      0.07     0.71     0.94        Inf         1    *
## ---
## 'CI': 90%-CI for one-sided and 95%-CI for two-sided hypotheses.
## '*': For one-sided hypotheses, the posterior probability exceeds 95%;
## for two-sided hypotheses, the value tested against lies outside the 95%-CI.
## Posterior probabilities of point hypotheses assume equal prior probabilities.

Conclusions:

• the probability of a positive effect of mussel clump area on number of individuals is 1. That is, there is, we are 100% confident that there is an effect.
• the evidence ratio is normally the ratio of the number of cases that satisty the hypothesis to the number of cases that do not. Since the number of cases that do not satisfy the hypothesis is 0, the evidence ratio is Inf (since division by 0)

Alternatively…

peake.brm4 %>% tidy_draws() %>% summarise(P=sum(b_logAREA>0)/n())

## # A tibble: 1 x 1
##       P
##   <dbl>
## 1     1

Conclusions:

• the probability of a positive effect of mussel clump area on number of individuals is 1. That is, there is, we are 100% confident that there is an effect.

Change 5000 to 10000

newdata = list(AREA=c(5000, 10000)) 
peake.brm4 %>% emmeans(~AREA,  at=newdata) %>% pairs()

##  contrast     estimate lower.HPD upper.HPD
##  5000 - 10000    -0.57    -0.672    -0.476
## 
## Point estimate displayed: median 
## Results are given on the log (not the response) scale. 
## HPD interval probability: 0.95

Conclusions:

• the change in expected number of individuals associated with an increase in mussel clump area from 5,000 to 10,000 is -0.57.

If we want to derive other properties, such as the percentage change, then we use tidy_draws() and then simple tidyverse spreadsheet operation.

peake.mcmc <- peake.brm4 %>% emmeans(~AREA,  at=newdata) %>% 
  tidy_draws() %>%
  rename_with(~str_replace(., 'AREA ', 'p')) %>%
  mutate(Eff=p10000 - p5000,
         PEff=100*Eff/p5000)
peake.mcmc %>% head

Now we can calculate the medians and HPD intervals of each column (and ignore the .chain, .iteration and .draw).

peake.mcmc %>% tidyMCMC(estimate.method='median',
                       conf.int=TRUE, conf.method='HPDinterval')

Alternatively, we could use median_hdci

peake.mcmc %>% median_hdci(PEff)

Conclusions:

• the percentage change in expected number of individuals associated with an increase in mussel clump area from 5,000 to 10,000 is 9.74% with a 95% credibility interval of 8.15 11.52

To get the probability that the effect is greater than a 50% increase.

peake.mcmc %>% summarise(P=sum(PEff>50)/n())

## # A tibble: 1 x 1
##       P
##   <dbl>
## 1     0

Conclusions:

• the probability that the number of individuals will increase by more than 50% following an increase in mussel clump area from 5,000 to 10,000 is

Finally, we could alternatively use hypothesis(). Note that when we do so, the estimate is the difference between the effect and the hypothesised effect (50%).

peake.mcmc %>% hypothesis('PEff>50')

## Hypothesis Tests for class :
##        Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio Post.Prob
## 1 (PEff)-(50) > 0   -40.26      0.84   -41.63   -38.85          0         0
##   Star
## 1     
## ---
## 'CI': 90%-CI for one-sided and 95%-CI for two-sided hypotheses.
## '*': For one-sided hypotheses, the posterior probability exceeds 95%;
## for two-sided hypotheses, the value tested against lies outside the 95%-CI.
## Posterior probabilities of point hypotheses assume equal prior probabilities.

Summary figure

rstanarm

## Using emmeans
peake.grid = with(peake, list(AREA = seq(min(AREA), max(AREA), len=100)))

newdata = emmeans(peake.rstanarm4, ~AREA, at=peake.grid, type='response') %>% as.data.frame
head(newdata)

ggplot(newdata, aes(y=prob, x=AREA)) + 
geom_point(data=peake, aes(y=INDIV)) +
geom_line() + 
geom_ribbon(aes(ymin=lower.HPD, ymax=upper.HPD), fill='blue', alpha=0.3) +
scale_y_continuous('Individuals') +
scale_x_continuous('Mussel clump area') +
    theme_classic()

brms

## Using emmeans
peake.grid = with(peake, list(AREA = seq(min(AREA), max(AREA), len=100)))

newdata = emmeans(peake.brm4, ~AREA, at=peake.grid, type='response') %>% as.data.frame
head(newdata)

ggplot(newdata, aes(y=prob, x=AREA)) + 
geom_point(data=peake, aes(y=INDIV)) +
geom_line() + 
geom_ribbon(aes(ymin=lower.HPD, ymax=upper.HPD), fill='blue', alpha=0.3) +
scale_y_continuous('Individuals') +
scale_x_continuous('Mussel clump area') +
    theme_classic()

References

Peake, A. J., and G. P. Quinn. 1993. “Temporal Variation in Species-Area Curves for Invertebrates in Clumps of an Intertidal Mussel.” Ecography 16: 269–77.